in terms of the hydraulic radius, RH, for the respective flows yield values

of EL/RHu, of 20.2 for the pipe and 5.9 for the infinitely wide channel.

Bowden (5) developed expressions for EL using several different velocity

and eddy diffusivity distributions with depth and showed that EL/u,h

ranged from 5.9 to 25 for the cases analyzed. Thus, the particular velocity

shear profile used along with the cross-sectional eddy diffusivity have

a significant effect on the magnitude of the predicted longitudinal

dispersion coefficient.

While the works of Taylor, Aris, and Elder proved the existence of

an effective dispersive mass transport mechanism and there was reasonable

agreement between laboratory experimental data for pipe and infinitely

wide open channel flow geometries, other investigators were finding that

observed values of the longitudinal dispersion coefficient for natural

rivers and streams were considerably higher than those predicted by theory.

As reported by Fischer (6) observed values for EL in natural watercourses

ranged from 50 to 700 hu, as compared with the 5.9 to 25 hu, range predicted

by Elder and Bowden. To account for this increased effect Fischer concluded

that the dominant mechanism in longitudinal dispersion was the interaction

of the velocity shear profile and turbulent mixing time scales across the

finite width of the channel as opposed to the vertical variations treated

by the previous investigators for infinitely wide channels. The effect

of asymmetrical flow cross sections on longitudinal dispersion was first

examined by Aris (3) who showed that the dispersion in a circular tube is

less than that in an elliptical one of the same area. This in addition to

the fact that all real watercourses have a finite width and that width-

to-depth ratios for natural channels are usually significantly greater

than unity makes Fischer's hypothesis very reasonable. For wide channels

the transverse mixing time, k /Ky, over some characteristic zs would tend

to be greater than the corresponding vertical mixing time scale, h2/Kz.

Fischer thus argued that the velocity shear profile across the channel

would have a greater longitudinal dispersive effect since an increased

mixing time would produce larger values of the spatial correlation u'c".

Using this as the basis for his analysis Fischer applied Taylor's technique

to a rectangular coordinate system and assumed that vertical variations

in c" were negligible compared to the transverse variations to obtain

u"- a (K (1-15)

where y is the coordinate axis across the channel.

Fischer then integrated (1-15) over the cross-sectional area of the

channel to obtain an expression for c" which he then correlated with u"

for the dispersive mass transport and using Taylor's definition of a

Fickian flux obtained a longitudinal dispersion coefficient of

w y
EL = q"(y)dy K hy T dy q"(y)dy (I-16)
0 0 0

where the depth, h, is a function of y; K is the transverse eddy diffusi-

vity; w is the width of the channel; and q"(y) is the depth integrated

flow defined by

h(y)

q"(y) = S u"(y,z) dz (1-17)
0

To obtain a more useful expression for EL Fischer defines a Lagrangian

time scale for cross-sectional mixing which he obtains through an extension

of Taylor's work on diffusion by continuous movements (7), and relates it

to the Eulerian time scale for cross-sectional mixing previously discussed.

Then, using Elder's (4) experimental determination that the transverse

eddy diffusivity in an infinitely wide channel with a steady unidirectional

flow can be expressed as

K = 0.23 hu, (1-18)

Fischer arrives at an alternate expression for EL given by

2
EL = 0.3 u" (1-19)
L RHU.

where s in a natural channel is the distance from the point of maximum
surface velocity to the most distant bank. To verify his theory Fischer
conducted a detailed set of laboratory experiments. Using width-to-depth
ratios ranging from 9 to 15.7 Fischer obtained good agreement between

predicted and observed values of EL using (1-16). However, (1-19) over-
predicted in each case, with a maximum error of 75 per cent.

Fischer (8) also applied his analysis to data obtained by other investigators

This case is comparable to Bowden's analysis for a very long period of
oscillation and supports his finding that Ex 1/2 EL for large T.
Holley et al. (11) successfully applied Taylor's original method of

analysis to obtain an analytical expression for T in a two dimensional
periodic, uniform shear flow. For their analysis they assumed the spatial

variation component of velocity to be

u" (z,t) = a z sin at (1-28)

where a is a constant. This is identical to the linear oscillatory flow

profile assumed by Okubo. This expression for u" is then used to force
the following form of the transport diffusion equation to obtain a solution

for c" (z,t)

(=) Kz, =-" (1-29)

The solutions obtained for u" and c" are then used to determine the depth

integrated time mean convective mass transport with respect to the depth
mean velocity. The resulting expression for the mean longitudinal dispersion
coefficient over the period of oscillation is
a2 T2 h4 1
S z n=l (2n-li)2{[ (2n-1)2 T' 2+

TK
where T' = -h. Holley et al. then proceed to relate T in an oscillating

flow to the dispersive process in an oscillating flow of infinite period

with the same velocity profile as given by (1-28). Solving for c" using

the steady state form of (1-29) and proceeding in the usual manner they

obtain an expression for T which they call E It is shown that E.

is one-half the corresponding value for EL in agreement with Bowden (5),

It should be noted that (III-9) implicitly satisfies the boundary condi-
h
tion at z = h since by definition S u"(z) dz = 0. Equation (III-9) is
0
then integrated once more over z and the constant of integration is

evaluated using the requirement that c"(z) must have a zero mean over
depth. The resulting solution for the concentration variation is then

c"(z) = K c z4 hz3 h2z2 h4
c"(z) ( 2 + l- (III-10)
c K 24 6 6 t5
zKz

Using (III-6) to substitute for K in (III-10) the expression for

c"(z) as a function of umax is

c"(z = a 4 hz3 h2z2 h'
c"(z) = max (T ) r-4- -6 + T- (III-ii)
K h2
z

III-A.3 Dispersive Mass Transport
The dispersive mass transport over the depth of flow is given by

S= S u"c"dz (III-12)

Substituting for u" and c" using (III-5) and (III-10) respectively,

(I1I-12) becomes

K2 aBE z h2 z4 hz3 h2z2 h
1 = -- () (2- hz + 3 24 6 6 45 dz
ezKZ 0

Carrying out the integration the mass transport is then

=(- ) 2K2h (III-13)
945EzKz

It shall be assumed for this analysis and for the analyses that

follow that Reynolds analogy relating the transfer of mass and momentum

by turbulent processes is applicable so that

E = K (III-14)
z z

This assumption is not necessary to obtain solutions for EL and Ex using

the method presented here; however as will be seen later its use simpli-

fies the forms of solutions considerably and facilitates interpretation

of the physical processes involved. Thus, with the aid of (III-14) and

the introduction of the vertical mixing time defined by

T h2 (III-15)
K
z

Equation (111-13) may be written as
2K2T3 h
S( z4 (III-16)

Assuming next that the required conditions for defining a longitudinal

dispersion coefficient as discussed in II-B are satisfied, then

m
L a)h
(- -

2K2T'
cz
EL =~9 (III-17)

Using (III-6) and (111-15) in (111-17) the longitudinal dispersion

coefficient as a function of uax is

8u2 T
max cz
EL ----4 (111-18)

The functional form of EL given by (111-18) is the same as that

obtained by Taylor for laminar unidirectional flow in a tube,(I-8). In

both cases the longitudinal dispersion varies directly as the product

of the square of the velocity and the cross-sectional mixing time.

Thus, for a given unidirectional velocity profile the longitudinal mass

transport of a substance increases linearly with the time required to

mix that substance over the flow cross section by diffusive processes.

III-B Oscillatory Flow

III-B.1 Velocity Distribution

The same assumptions used in III-A regarding selection of coordinate

system, and the use of the Boussinesq approximation for expressing the

viscous term in the equation of motion will be applied here. The govern-

with only the real part of (111-23) having any physical meaning. Its
form is clearly that of a damped progressive shear wave propagating

upward through the water column and causing a temporal phase shift in

the velocity as a function of z. It was also pointed out by Segall and

Gidlund (20) that the solution as expressed by (111-23) correctly predicts

flow reversals in the lower momentum layers of the fluid near the bed

prior to a shift in flow direction of the higher momentum layers near

the free surface. These effects are extremely important to the disper-

sive transport mechanism.

The spatial variation of the velocity, u"(z,t), is obtained by

averaging (111-23) over depth and subtracting out the mean from the

total velocity according to
u"(z,t) = u(z,t) u(t)

The result is then

u"(z,t) = iKeiot { sinh (l+i)h cosh B(1+i)(h-z)} (111-25)
a cosh B(1+i)h (l+i)h
For reasons that will become obvious in the next section the velocity

variation solution as given by (111-25) will now be expanded in a Fourier

cosine series of the form

u"(z,t) = an cos nT eiat (III-26)
n=l

where coefficients an are complex. The a0 term in the series has been
omitted to satisfy the zero mean requirement for u". It is noted by
Hildebrand (22) that any piecewise differentiable function may be com-
pletely represented by a cosine series of the form a cos z- over the
closed interval 0 < z < h. Proceeding then,
h
S2 iK sinh B(1+i)h nrz d
a h (1+i)h cosh B(1+i)h cos dz
0
h
2 ( iK niz
h a cosh (1+i)h cosh B(1+i)(h-z) cos n dz
o
The first term on the right hand side integrates to zero while the second
term must be integrated by parts twice which yields for an,
2K Bh(1-i) sinh B(l+i)h
n o cosh B(1+i)h [(ni)2 + i2(Bh)2] (III-27)

Since only the real part of u"(z,t) is of any interest, (III-27) will now
be put in polar form for incorporation into (III-26). Considering each
complex term separately,

umax, of the periodic velocity function and the surface excursion length,
L, experienced by a particle during one-half of the period of oscilla-
tion are obtained by solving for K as a function of umax and L using
max

(III-23). Details of this development are presented in Appendix A, the
results of which are

III-B.2 Concentration Distribution
For temporally periodic u and c, uniform flow, and constant eddy
diffusivity it was shown in II-C that the transport diffusion equation
as described from a coordinate system moving with the cross-sectional
mean velocity could be written as

ac" K c" = -u"c (111-45)
3t z azz 'l

where the rate of change with respect to C will hereafter be implied.

The concentration variation, c"(z,t), shall be assumed to be of the form

iat cos mnz iat
c"(z,t) = c1(z)eit a cos eit (III-46)
m=l

where a' is complex. This expression for c" is seen to satisfy the require-

ments of periodicity and zero diffusive flux of substance across the flow

boundaries. Moreover, if (III-46) is substituted into (III-45) and

operated on, the left hand side is composed of cos nz- terms only. Thus,

to satisfy the conditions of equality, the expansion of (III-25) for u"

which due to the orthogonal properties of cos mT and cos n requires
h d cos requires
that m=n for a non-trivial solution. The Fourier coefficient a' can
m
then be written as
S h2a
a' m ( m (III-48)
SKz (mr)2 + i(h2)]
Kz

The required solution for c"(z,t) is arrived at by: (1) using

(III-26) and (III-37) to determine the polar form of am; (2) transforming

(nm)2 (III-57)
[(nr)4 + (2Tz)22
n=1
Equation (111-56) is considered to be the most useful and descrip-
tive of the solutions presented for T in two dimensional oscillatory
shear flow. Therefore, to facilitate a discussion of these results in

III-C Discussion of Two Dimensional Shear Flow Results
III-C.1 Dispersion in Unidirectional Flow as a Limit of the Oscillatory
Flow Case
This discussion shall begin by relating the oscillatory flow disper-
sion coefficient, T to its "corresponding" unidirectional flow
coefficient, EL, as has been previously done by Holley et al., Awaya,
and Fukuoka; and shall then proceed to show how the unidirectional process
is in fact a limiting case of an oscillatory process which has its own
distinct characteristics. Thus, solutions obtained in III-B.3 forT as a function of the pressure gradient and the surface amplitude
of the velocity are normalized by the "corresponding" solutions for EL
obtained in III-A.3. Dividing (111-55) by (111-17) and (111-56) by
(III-18) yields

us(y,z) a' cos (1 (n 2m+1 1) y (IV-4)
m=0 n=O
is assumed. An examination of (IV-4) with the statement of the boundary
value problem above shows that the assumed solution satisfies the no
slip conditions on the perimeter of the section. Moreover, (IV-4) also
satisfies the condition of zero vertical shear at the free surface, and
zero lateral shear at the centerline of the channel. The assumed solution
is then substituted into (IV-3) and the Fourier coefficients, amn, are
determined in the usual manner through the use of the orthogonal proper-
ties of the cosine function. Details of this analysis are presented in
Appendix B. Solving for amn and thus determining the form of us(y,z),
the complete solution for the velocity function u(y,z,t) using (IV-4)
and (IV-2) is then

Since only the real portions of these expressions have any physical
meaning, (IV-7) is transformed to polar form as was done in Chapter III
for the two dimensional case. In addition, it will be assumed once again
for simplification of results that Reynolds analogy applies thereby
allowing the introduction of the vertical mixing time Tcz and the defini-
tion of a comparable lateral mixing time as

T = (/2)2
cy K
y

The solution for u"(y,z,t) as a function of the pressure
then be written as
u(yzt) 8K ()m+n ei(ct-ai(m,n))
u"(y,z,t) = 7-.)
m = 0 n = 0 (2 m+1)(2 n+l) R0 (m,n)

In order to put (IV-11) in a more usable form, (IV-5) is used to
obtain a relationship between the pressure gradient, K, and umax, where
umax is now defined as the amplitude of the periodic velocity function at
the free surface centerline. Thus, from (IV-5)

u(0,0,t) = .
m = 0 n= 0

-16 K(-)m+ ei (-14)
T2(2 m+l)(2 n+1) {Ez[2 m+) 2+ [(2 n+l)l 2 i w

which upon maximization with repsect to time yields the relationship
umaxir3
K -max (IV-15)
8T

where,

S = ,-)m+n(R(m,n)-l)
m 0 n 0 (2 m+l)(2 n+l)Rl(m,n

+ (-)m+n (IV-16)
m = 0 n = (2 m+l)(2 n+l) R(m,n)

The details of developing (IV-15) from (IV-14) are presented in Appendix
C. Substitution of (IV-15) into (IV-11) for K yields the following
expression for u" as a function of umax
metUm)max)

u"(y,z,t) umax (-Im+ni(t-(mn))
m = 0 n = 0 (2 m+1)(2 n+l)R (m,n)

cos (2 m+) cos (2 n+l)y g(m,n) (IV-17)
2h w

IV-A.2 Concentration Distribution

The form of the transport diffusion equation used to solve for

c"(y,z,t) is identical to (11-27), or

c" a2cl' a2c a3 2
K z K = u" (IV-18)
at z z2 Y y2 aT

where u"(y,z,t) is given by (IV-7). It is assumed that c"(y,z,t) is of

the form

c"(y,z,t) = c" (y,z) eiot (IV-19)
s

where c"(y,z) is complex following the same approach used with u"(y,z,t).

Substituting (IV-19) into (IV-18) and requiring zero diffusive flux across

where amn is given by (IV-9). The unknown Fourier coefficients ask are
solved for in the same manner as was used to determine amn. The detailed
mathematics of the solution are presented in Appendix B, the result of
which is given by

at 256K 2i (IV-23)
aik = 4{_- 2 -) 2 O 2
Kz() y+ K( + i p = 0 q = 0

(-1)-(P+q)
[(2 p+1) -(2Z)2][(2 q+l)2-(2k)2]{[E (2 ]2 r(2 q+1z12 + io}

Thus, the solution for c"(y,z,t) includes four infinite summations
arising from the non-orthogonal nature of the cosine functions
representing u"(y,z) and c'(y,z). Again introducing Reynolds analogy
and the expression for the vertical and transverse mixing times, the
solution for c"(y,z,t) in polar form as a function of the pressure
gradient is

The factorPZik is necessarily included in (IV-24) to properly account
for the a6k and a'o terms, and to insure that c"(y,z,t) has a zero mean
over the flow cross-sectional area. Detailed mathematics for arriving
at (IV-24) are given in Appendix B.
The expression of c"(y,z,t) as a function of umax is determined by
substitution for K in (IV-24) using (IV-15) to yield,